Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received August 6, 2013; final manuscript received February 3, 2014; published online April 3, 2014. Assoc. Editor: Philippe Wenger.

Abstract

When large surfaces need to be covered by a robotic system, the most common solution is to design or employ a robot with a comparably large workspace (WS), with high costs and high power requirements. In this paper, we propose a new methodology consisting in an efficient partitioning of the surface, in order to use robotic systems with a workspace of arbitrarily smaller size. These robots are called repetitive workspace robots (RWR). To support this method, we formally define a general index IRWR in order to evaluate the covering efficiency of the workspace. Three algorithms to compute the index are presented, the uniform expansion covering algorithm (UECA), the corrected inertial ellipsoid covering algorithm (CIECA), and the genetic covering algorithm (GCA). The GCA, which delivers a solution in the proximity of the global-best one, is used as a baseline for a comparison between the UECA and the CIECA. Eventually, we present the results of a performance analysis of the three algorithms in terms of computing time. The results show that the CIECA is the best algorithm for the evaluation of the IRWR, almost reaching the global-best solutions of the GCA. Finally, we illustrate a practical application with a comparison between two commercial industrial paint robots: the ABB™ IRB 550 and the CMA® Robotics GR 6100.

Figures

A repetitive workspace robot used in the painting process of a large building. The elevator base moves left, right, up and down, then holding the position, it paints the wall contained in the local workspace.

Geometrical representation of the rotation and translation of the generic workspace W. In (a), ξ, ζ are defined as in Fig. 4 and define the primitive cell of points i, j, k. Furthermore, x, y are the unit vectors of the original coordinate system X, Y rotated by an angular value of γ; O is the origin and is located on the middle point of the link between i and k. The translation vector vt is visible, along with its unit vector v∧t and module Δs. In (b) the translation Tvt is applied to a point p ∈ W, resulting in the point q ∈ W˜. In (c), the translation of the workspace W into four copies W˜1, W˜2, W˜3, and W˜4, is presented, with Δx1, Δx2, Δy the scalar quantities that define the vectors vt,1,…, vt,4. In (d), we can see that the W˜ workspaces are thus arranged according to a lattice of primitive cell i, j, k.

Covering efficiency of the working area (dashed line) when local workspaces are arranged in a lattice. The covered area is shown in grey. In (a), a square and a round workspace are shown; these have the same area. In (b), one can see a partial covering of the working area resulting from the use of the square workspace, whereas in (c), the same is visible in case the circular one is employed.

Examples of the covering problem. In (a), a classic SCP is illustrated, where A (bold line) is a surface of finite extension, and B1, B2, …, Bi are repetitions of a finite surface B. In (b), a nonstandard SCP is shown, where the surface A has infinite extension and B is tiled evenly according to a lattice structure. The dashed line shows a finite partition of A.

Three general types of domains are presented. In (a), a connected and island-free space is shown, in (b), an island-free but nonconnected domain is visible, and in (c), a connected space with an island is illustrated. Note that in (b) and (c), more than one boundary is present, while in (a) there is a single one.

Translation of the original workspace and expansion of the union of the resulting areas. The arrows show the translation of the W˜ areas. The grey area shows the expanded area Wexp. In (a), the robot workspace W is shown. In (b), a general translation step is illustrated, and in (c), the maximum expansion step is appreciable, with the presence of a gap at the center of the frame; this is representative of the criterion for the maximum translation (expansion) of the surface.

An illustration of a “sock anomaly.” Workspaces with shapes similar to the one in (a) can result in the nondetection of the islands visible in (d) as dashed lines, if UECA were to be used. Indeed, if the expansion in (b) and in (c) is examined, the criterion nb = 1 is still verified.

Calculation of the inertia ellipsoid with the aid of a rotating axis g centered in the barycenter Ω. In (a), the 2D body W is shown. A point t ∈ W is also highlighted, as are the x, y coordinate system and axis g. For a specified angle θ of said axis, the minimum distance d of point t from the axis itself can be easily computed. In (b), the resulting 2D inertial ellipse is visible, in the (r, θ) coordinates.

Different types of Ig computed for two different shapes, a square, and an oblique rectangle. In the second column, Ig(θ) is visible: This is the distribution of the moment of inertia along a rotating axis. In the third column, Ig'(θ), an adjusted, third degree moment of inertia is visible, and in the fourth column, Ig"(θ) is presented, which is a first-degree-adjusted moment of inertia. It is important to note that Ig(θ) in the first column is what leads to the usual inertia ellipsoid.

Results summary for the three methodologies to calculate the IRWR. In the first column, the shape of the workspace is visible (letters a–h). In columns two, four, and six, the configuration of maximum expansion is shown for the related shape, respectively, for GCA, UECA, and CIECA. Finally, in columns three, five, and seven, the actual associated index IRWR is reported. Black areas are considered as workspace, whereas the grey ones are intersections between overlapping expanded workspaces.

Maximum sagittal workspaces and workspace bulk schematics of two industrial spray-paint robots. In (a), the ABB™ IRB 550 is shown and in (b), the CMA® Robotics GR 6100 is represented. Dimensions are in mm.

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